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Dans cette thèse, nous nous intéressons à deux approximations hyperboliques des équations de Navier-Stokes incompressibles en dimensions 2 et 3 d'espace. Dans un premier temps, on considère une perturbation hyperbolique de l'équation de la chaleur, introduite par Cattaneo en 1949, pour remédier au paradoxe de la propagation instantanée de cette équation. En 2004, Brenier, Natalini et Puel remarquent que la même perturbation, qui consiste à rajouter ε∂tt à l'équation, intervient en relaxant les équations d'Euler. En dimension 2, les auteurs montrent que, pour des sonnées régulières et sous certaines hypothèses de petitesse, la solution globale de la perturbation converge vers l'unique solution globale de (NS). En 2007, Paicu et Raugel améliorent les résultats de [BNP] en étendant la théorie à la dimension 3 et en prenant des données beaucoup moins régulières. Nous avons obtenu des résultats de convergence, avec données de régularité quasi-critique, qui complètent et prolongent ceux de [BNP] et [PR]. La seconde approximation que l'on considère est un nouveau modèle hyperbolique à vitesse de propagation finie. Ce modèle est obtenu en pénalisant la contrainte d'incompressibilité dans la perturbation de Cattaneo. Nous démontrons que les résultats d'existence globale et de convergence du précédent modèle sont encore vérifiés pour celui-ci. / In this work, we are interested in two hyperbolic approximations of the 2D and 3D Navier-Stokes equations. The first model we consider comes from Cattaneo's hyperbolic perturbation of the heat equation to obtain a finite speed of propagation equation. Brenier, Natalini and Puel studied the same perturbation as a relaxed version of the 2D Euler equations and proved that the solution to this relaxation converges towards the solution to (NS) with smooth data, provided some smallness assumptions. Later, Paicu and Raugel improved their results, extending the theory to the 3D setting and requiring significantly less regular data. Following [BNP] and [PR], we prove global existence and convergence results with quasi-critical regularity assumptions on the initial data. In the second part, we introduce a new hyperbolic model with finite speed of propagation, obtained by penalizing the incompressibility constraint in Cattaneo's perturbation. We prove that the same global existence and convergence results hold for this model as well as for the first one.

Turbulent flows occur in various fields and are a central, yet an extremely complex, topic in fluid dynamics. Understanding the behaviour of fluids is vital for multiple research areas including, but not limited to: biological models, weather prediction, and engineering design models for automobiles and aircraft. In this thesis, we study a number of fundamental problems that arise in 2D turbulent flows, using the 2D Navier-Stokes system. Introducing optimization techniques for systems described by partial differential equations (PDE), we frame these problems such that they can be solved using computational methods. We utilize adjoint calculus to build the computational framework to be implemented in an iterative gradient flow procedure, using the "optimize-then-discretize" approach. Pseudospectral methods are employed for solving PDEs in a numerically efficient manner. The use of optimization methods together with computational mathematics in this work provides an illuminating perspective on fluid mechanics. We first apply these techniques to better understand enstrophy dissipation in 2D Navier-Stokes flows, in the limit of vanishing viscosity. By defining an optimization problem to determine optimal initial conditions, multiple branches of local maximizers were obtained each corresponding to a different mechanism producing maximum enstrophy dissipation. Viewing this quantity as a function of viscosity revealed quantitative agreement with an analytic bound, demonstrating the sharpness of this bound. We also introduce an extension of this problem, where enstrophy dissipation is maximized in the context of kinetic theory using the Boltzmann equation. Secondly, these PDE-constrained optimization techniques were used to probe the fundamental limitations on the performance of the Leith eddy-viscosity closure model for 2D Large-Eddy Simulations of the Navier-Stokes system. Obtained by solving an optimization problem with a non-standard structure, the results demonstrate the optimal eddy viscosities do not converge to a well-defined limit as regularization and discretization parameters are refined, hence the problem of determining an optimal eddy viscosity is ill-posed. Further extending the problem of finding optimal eddy-viscosity closures, we consider imposing an additional nonlinear constraint on the control variable in the problem, in the form of requiring the time-averaged enstrophy be preserved. To address this problem in a novel way, we employ adjoint calculus to characterize a subspace tangent to the constraint manifold, which allows one to approximately enforce the constraint. Not only do we demonstrate that this produces better results when compared to the case without constraints, but this also provides a flexible computational framework for approximate enforcement of general nonlinear constraints. Lastly in this thesis, we introduce an optimization problem to study the Kolmogorov-Richardson energy cascade, where a pathway towards solutions is outlined. / Thesis / Doctor of Philosophy (PhD)

One of the most prominent open problems in mathematical physics is determining whether solutions to the incompressible three-dimensional (3D) Navier-Stokes system, corresponding to arbitrarily large smooth initial data, remain regular for arbitrarily long times. A promising approach to this problem relies on the fact that both the smoothness of classical solutions and the uniqueness of weak solutions in 3D flows are ultimately controlled by the growth properties of the $H^1$ seminorm of the velocity field U, also known as the enstrophy. In this context, the sharpness of analytic estimates for the instantaneous rate of growth of the $H^2$ seminorm of U in two-dimensional (2D) flows, also known as palinstrophy, and for the instantaneous rate of growth of enstrophy in 3D flows, is assessed by numerically solving suitable constrained optimization problems. It is found that the instantaneous estimates for both 2D and 3D flows are saturated by highly localized vortex structures. Moreover, finite-time estimates for the total growth of palinstrophy in 2D and enstrophy in 3D are obtained from the corresponding instantaneous estimates and, by using the (instantaneously) optimal vortex structures as initial conditions in the Navier-Stokes system and numerically computing their time evolution, the finite-time estimates are found to be uniformly sharp for 2D flows, and sharp over increasingly short time intervals for 3D flows. Although computational in essence, these results indicate a possible route for finding an extreme initial condition for the Navier-Stokes system that could lead to the formation of a singularity in finite time. / Thesis / Doctor of Philosophy (PhD)

This thesis derives a first approximation model that will describe the flow of fluid in microfluidic devices such as in microchannels, microdiffusers and micronozzles using electrical network modelling. The important parameter that is of concern is the flow rates of these devices. The purpose of this work is to contribute to the physical component of our interactive Virtual Reality (VR)-prototyping tool for MEMS, with emphasis on fast calculations for interactive CAD design. Current calculations are too time consuming and not suitable for interactive CAD with dynamic animations. This work contributes to and fills the need for the development of MEMS dynamic visualisation, showing the movement of fluid within microdevices in time scale. Microfluidic MEMS devices are used in a wide range of applications, such as in chemical analysis, gene expression analysis, electronic cooling system and inkjet printers. Their success lies in their microdimensions, enabling the creation of systems that are considerably minute yet can contain many complex subsystems. With this reduction in size, the advantages of requiring less material for analysis, less power consumption, less wastage and an increase in portability becomes their selling point. Market size is in excess of US$50 billion in 2004, according to a study made by Nexus. New applications are constantly being developed leading to creation of new devices, such as the DNA and the protein chip. Applications are found in pharmaceuticals, diagnostic, biotechnology and the food industry. An example is the outcome of the mapping and sequencing of the human genome DNA in the late 1990's leading to greater understanding of our genetic makeup. Armed with this knowledge, doctors will be able to treat diseases that were deemed untreatable before, such as diabetes or cancer. Among the tools with which that can be achieved include the DNA chip which is used to analyse an individual's genetic makeup and the Gene chip used in the study of cancer. With this burgeoning influx of new devices and an increase in demand for them there is a need for better and more efficient designs. The MEMS design process is time consuming and costly. Many calculations rely on Finite Element Analysis, which has slow and time consuming algorithms, that make interactive CAD unworkable. This is because the iterative algorithms for calculating the animated images showing the ongoing proccess as they occur, are too slow. Faster computers do not solve the void of efficient algorithms, because with faster computer also comes the demand for a fasters response. A 40 - 90 minute FEA calculation will not be replaced by a faster computer in the next decades to an almost instant response. Efficient design tools are required to shorten this process. These interactive CAD tools need to be able to give quick yet accurate results. Current CAD tools involve time consuming numerical analysis technique which requires hours of numerous iterations for the device structure design followed by more calculations to achieve the required output specification. Although there is a need for a detailed analysis, especially in solving for a particular aspect of the design, having a tool to quickly get a first approximation will greatly shorten the guesswork involved in determining the overall requirement. The underlying theory for the fluid flow model is based on traditional continuum theory and the Navier-Stokes equation is used in the derivation of a layered flow model in which the flow region is segmented into layered sections, each having different flow rates. The flow characteristics of each sections are modeled as electrical components in an electrical circuit. Matlab 6.5 (MatlabTM) is used for the modelling aspect and Simulink is used for the simulation.

Approximation and interpolation employing radial basis functions has found important applications since the early 1980's in areas such as signal processing, medical imaging, as well as neural networks. Several applications demand that certain physical properties be fulfilled, such as a function being divergence free. No such class of radial basis functions that reflects these physical properties was known until 1994, when Narcowich and Ward introduced a family of matrix-valued radial basis functions that are divergence free. They also obtained error bounds and stability estimates for interpolation by means of these functions. These divergence-free functions are very smooth, and have unbounded support. In this thesis we introduce a new class of matrix-valued radial basis functions that are divergence free as well as compactly supported. This leads to the possibility of applying fast solvers for inverting interpolation matrices, as these matrices are not only symmetric and positive definite, but also sparse because of this compact support. We develop error bounds and stability estimates which hold for a broad class of functions. We conclude with applications to the numerical solution of the Navier-Stokes equation for certain incompressible fluid flows.

Um problema auxiliar crucial à análise do problema de Stokes Compressível é estudado via a técnica de potenciais de camada dupla em regiões Lipschitz através de um método primeiro utilizado por Verchota e subseqüentemente estendido ao caso parabólico por Brown e Shen. Desse modo, mediante a utilização e cálculo da condição de salto na fronteira é possível estabelecer a existência e unicidade da solução em apropriados espaços funcionais via o estudo de potenciais de camada. / An auxiliary problem crucial to the analysis of the compressible Stokes problem is studied by means of the technique of double layer in Lipschitz regions through a method first used by Verchota and subsequently extended to the parabolic case by Brown and Shen. In this way through the use and calculation of the boundary jump condition it is possible to establish the existence and unicity of the solution in appropriate function spaces via the study of boundary layer potentials.

Um problema auxiliar crucial à análise do problema de Stokes Compressível é estudado via a técnica de potenciais de camada dupla em regiões Lipschitz através de um método primeiro utilizado por Verchota e subseqüentemente estendido ao caso parabólico por Brown e Shen. Desse modo, mediante a utilização e cálculo da condição de salto na fronteira é possível estabelecer a existência e unicidade da solução em apropriados espaços funcionais via o estudo de potenciais de camada. / An auxiliary problem crucial to the analysis of the compressible Stokes problem is studied by means of the technique of double layer in Lipschitz regions through a method first used by Verchota and subsequently extended to the parabolic case by Brown and Shen. In this way through the use and calculation of the boundary jump condition it is possible to establish the existence and unicity of the solution in appropriate function spaces via the study of boundary layer potentials.

Um problema auxiliar crucial à análise do problema de Stokes Compressível é estudado via a técnica de potenciais de camada dupla em regiões Lipschitz através de um método primeiro utilizado por Verchota e subseqüentemente estendido ao caso parabólico por Brown e Shen. Desse modo, mediante a utilização e cálculo da condição de salto na fronteira é possível estabelecer a existência e unicidade da solução em apropriados espaços funcionais via o estudo de potenciais de camada. / An auxiliary problem crucial to the analysis of the compressible Stokes problem is studied by means of the technique of double layer in Lipschitz regions through a method first used by Verchota and subsequently extended to the parabolic case by Brown and Shen. In this way through the use and calculation of the boundary jump condition it is possible to establish the existence and unicity of the solution in appropriate function spaces via the study of boundary layer potentials.

We examine the Rayleigh–Bénard convection as modelled by the Boussinesq equation. Our aim is at deriving bounds for the heat enhancement factor in the vertical direction, the Nusselt number, which reproduce physical scalings. In the first part of the dissertation, we examine the the simpler model when the acceleration of the fluid is neglected (Pr=∞) and prove the non-optimality of the temperature background field method by showing a lower bound for the Nusselt number associated to it. In the second part we consider the full model (Pr<∞) and we prove a new upper bound which improve the existing ones (for large Pr numbers) and catches a transition at Pr~Ra^(1/3).